2.1 Measures

In the spirit of Bourbacki [Reference Bourbaki9], by a measure, we understand a linear functional on $C_{\mathsf {c}}(G)$ which is continuous with respect to the inductive topology. This notion corresponds to the classical concept of a Radon measure (see [Reference Richard and Strungaru35, Appendix]). For the case $G={\mathbb R}^d$ , a clear exposition of this is given in [Reference Baake and Grimm3].

Definition 2.1 A linear functional $\mu : C_{\mathsf {c}}(G) \to {\mathbb C}$ is called a Radon measure (or simply a measure) if for each compact set $K \subseteq G$ there exists a constant $C_K$ such that, for all $\varphi \in C_{\mathsf {c}}(G)$ with $\mbox {supp}(\varphi ) \subseteq K$ , we have

$$ \begin{align*} \left| \mu(\varphi) \right| \leq C_K \| \varphi \|_{\infty} \,. \end{align*} $$

We will often write $\int _G \varphi (t) d \mu (t)$ instead of $\mu (\varphi )$ .

A measure $\mu $ is called positive if for all $\varphi \in C_{\mathsf {c}}(G)$ with $\varphi \geq 0$ we have $\mu (\varphi ) \geq 0$ .

By the Riesz representation theorem [Reference Rudin37], a positive Radon measure is simply a positive regular Borel measure. Moreover, each Radon measure is a linear combination of (at most four) positive Radon measures [Reference Richard and Strungaru35, Appendix].

Next, we review the total variation of a measure.

Definition 2.2 Given a measure $\mu $ , we can define [Reference Pedersen32, Reference Reiter and Stegeman33, Reference Richard and Strungaru35] a positive measure $\left | \mu \right |$ , called the total variation of $\mu $ , such that, for all $\varphi \in C_{\mathsf {c}}(G)$ with $\varphi \geq 0$ , we have

$$ \begin{align*} \left| \mu \right| (\varphi) = \sup \{ \left|\mu(\psi) \right| : \psi \in C_{\mathsf{c}}(G), \mbox{ with } |\psi| \leq \varphi \} \,. \end{align*} $$

We are now ready to introduce the concept of translation boundedness for measures and norm almost periodicity.

Definition 2.3 Let $A \subseteq G$ be a fixed precompact set with nonempty interior. We define the A-norm of $\mu $ via

$$ \begin{align*} \| \mu \|_A := \sup_{x \in G} \left| \mu \right| (x+A) \,. \end{align*} $$

A measure $\mu $ is called translation bounded if $\| \mu \|_A <\infty $ .

Remark 2.4 ([Reference Baake and Moody7, Reference Spindeler and Strungaru42])

Different precompact sets $A_1, A_2$ with nonempty interior define equivalent norms. Therefore, the definition of translation boundedness does not depend on the choice of A.

This allows us to define

$$ \begin{align*} {\mathcal M}^{\infty}(G) := \{ \mu : \mu \mbox{ is a translation bounded measure} \} \,. \end{align*} $$

Then $({\mathcal M}^{\infty }(G), \|. \|_A)$ is a normed space. It is in fact a Banach space [Reference Richard and Strungaru36].

Next, we review the definition of norm almost periodicity as introduced in [Reference Baake and Moody7].

Definition 2.5 Let $ A \subseteq G$ be a fixed precompact set with nonempty interior. A measure $\mu \in {\mathcal M}^{\infty }(G)$ is called norm almost periodic if, for each $\varepsilon>0$ , the set

$$ \begin{align*} P_{\varepsilon}^A (\mu) := \{ t \in G : \| T_t \mu -\mu \|_A < \varepsilon \} \, \end{align*} $$

of $\varepsilon $ -norm almost periods of $\mu $ is relatively dense.

As discussed above, different precompact sets define equivalent norms. This means that while the set of $\varepsilon $ -norm almost periods on $\mu $ depends on the choice of A, the almost periodicity of $\mu $ is independent of this choice.

Any norm almost periodic measure is strongly almost periodic [Reference Baake and Moody7], and the two concepts are equivalent for measures with Meyer set support [Reference Baake and Moody7]. In general, norm almost periodicity is an uniform version of strong almost periodicity [Reference Spindeler and Strungaru42, Theorem 4.7]. The class of norm almost periodic pure point measure was studied in detail and characterized in [Reference Strungaru, Baake and Grimm46].

Let us next recall positive-definiteness for functions and measures. For more details, we recommend [Reference Berg and Forst8, Reference Moody, Strungaru, Baake and Grimm31].

Definition 2.6 A function $f : G\longrightarrow {\mathbb C}$ is called positive-definite if, for all $ n \in \mathbb N$ and all $x_1,\ldots , x_n\in G$ , the matrix $\left (f (x_k - x_l)\right )_{k,l=1,\ldots , n}$ is positive Hermitian. This is equivalent to

$$\begin{align*}\sum_{k,l=1}^n \overline{c_l} f (x_k - x_l) c_k \geq 0 \qquad \forall n \in \mathbb N, x_1,\ldots, x_n\in G, c_1, \ldots , c_n \in {\mathbb C} \,. \end{align*}$$

A measure $\mu $ is called positive-definite if, for all $ \varphi \in C_{\mathsf {c}}(G),$ we have

$$ \begin{align*} \mu (\varphi * \widetilde{\varphi}) \geq 0 \,.\end{align*} $$

This is equivalent to $\mu *\varphi *\tilde {\varphi }$ being a positive-definite function for all $\varphi \in C_{\mathsf {c}}(G)$ [Reference Berg and Forst8, Reference Moody, Strungaru, Baake and Grimm31].

We complete the subsection by reviewing the notion of Fourier transformability for measures. For a more detailed review of the subject, we recommend [Reference Moody, Strungaru, Baake and Grimm31].

Definition 2.7 A measure $\mu $ on G is called Fourier transformable if there exists a measure $\widehat {\mu }$ on $\widehat {G}$ such that, for all $\varphi \in K_2(G)$ , we have $|\check {\varphi }| \in L^1( |\widehat {\mu } |)$ and

2.2 Cut-and-project schemes and Meyer sets

In this part, we review some notions related to the cut-and-project formalism. For more details, we recommend [Reference Baake and Grimm3, Reference Moody and Moody28, Reference Moody, Axel, Dénoyer and Gazeau29].

Given a CPS $(G,H, {\mathcal L})$ , we will denote by $L:= \pi _G({\mathcal L})$ . Then, $\pi _G$ induces a group isomorphism between ${\mathcal L}$ and L. Composing the inverse of this with the second projection $\pi _H$ , we get a mapping

$$ \begin{align*} \star : L \to H, \end{align*} $$

which we will call the $\star $ -mapping. We then have

$$ \begin{align*} {\mathcal L} = \{ (x,x^{\star}) : x \in L \} \,. \end{align*} $$

Given a CPS $(G, H, {\mathcal L})$ and a subset $W \subseteq H$ we can define

$$\begin{align*}{\LARGE \curlywedge}(W):= \{ x \in L : x^{\star} \in W \} \,. \end{align*}$$

When W is precompact, we will call ${\LARGE \curlywedge }(W)$ a weak model set. If furthermore W has nonempty interior ${\LARGE \curlywedge }(W)$ is called a model set.

Next, let us review the concept of a Meyer set, which plays a fundamental role in this paper.

For equivalent characterizations of Meyer sets, see [Reference Lagarias18, Reference Lagarias, Baake and Moody19, Reference Meyer26, Reference Moody and Moody28, Reference Strungaru, Baake and Grimm46]. Of importance to us will be the following result.

We should emphasize here that the key for all results below is that fact that a Meyer set is a subset of a model set, and relative denseness plays no role. Because of this, in the spirit of [Reference Strungaru49], we will refer to an arbitrary subset of a (weak) model set as a weak Meyer set. It is obvious that a subset of a weak Meyer set is a weak Meyer set and that a measure is supported inside a Meyer set if and only if its support is a weak Meyer set.

Given a CPS $(G,H, {\mathcal L})$ , the map

(2.1) $$ \begin{align} L \ni x \to (x,x^{\star}) \in {\mathcal L} \, \end{align} $$

is a group isomorphism, and hence it induces an isomorphism between the spaces of (bounded) functions on L and ${\mathcal L}$ , respectively. Since ${\mathcal L}$ is a discrete group, the space of (translation bounded) measures on ${\mathcal L}$ can be identified with the space of (bounded) functions on ${\mathcal L}$ . On another hand, L is typically dense in G, and many functions on L do not induce pure point measures on G.

For us, of interest will be measures supported inside weak model sets ${\LARGE \curlywedge }(W)$ . Since ${\LARGE \curlywedge }(W)$ is uniformly discrete [Reference Moody and Moody28], the space of (translation bounded) measures on ${\LARGE \curlywedge }(W)$ can be identified with the space of (bounded) functions on ${\LARGE \curlywedge }(W)$ , and corresponds via the above isomorphism with the spaces of (translation bounded) measures or (bounded) functions on ${\mathcal L}$ , respectively, that are supported inside $G \times W$ .

Our focus in this paper is on these two spaces. We will study them as spaces of measures, and we will be interested in the relation between the Fourier theory of these two spaces, and the behavior of the Fourier transform with respect to the isomorphism induced by (2.1). For this reason, let us introduce the following notations.

Given a CPS $(G,H, {\mathcal L})$ and a compact set W, we denote by

$$ \begin{align*} {\mathcal M}^{\infty}({\LARGE \curlywedge}(W)) &:= \{ \mu \in {\mathcal M}^{\infty}(G) : \mbox{supp}(\mu) \subseteq {\LARGE \curlywedge}(W)\} \,;\\ {\mathcal M}^{\infty}({\mathcal L}; W) &:=\{ \nu \in {\mathcal M}^{\infty}(G \times H) : \mbox{supp}(\nu) \subseteq \left( {\mathcal L} \cap (G \times W) \right) \} \,. \end{align*} $$

The isomorphism (2.1) induces a bijection $f: {\LARGE \curlywedge }(W) \to {\mathcal L} \cap (G \times W)$ . This induces a bijective map $\mathbb {L}_{G,H,{\mathcal L}, W}: {\mathcal M}^{\infty }({\LARGE \curlywedge }(W)) \to {\mathcal M}^{\infty }({\mathcal L}; W)$ , taking a measure on ${\LARGE \curlywedge }(W)$ into its pushforward via f, defined by

$$\begin{align*}\mathbb{L}_{G,H,{\mathcal L}, W}(\mu)= \sum_{(x,x^{\star}) \in {\mathcal L}} \mu(\{x \}) \delta_{(x,x^{\star})} \,, \end{align*}$$

with inverse $\mathbb {P}_{G,H,{\mathcal L}, W}: {\mathcal M}^{\infty }({\mathcal L} ;W) \to {\mathcal M}^{\infty }({\LARGE \curlywedge }(W))$

$$\begin{align*}\mathbb{P}_{G,H,{\mathcal L}, W}(\nu)= \sum_{(x,x^{\star}) \in {\mathcal L}} \nu(\{(x,x^{\star}) \}) \delta_{x} \,. \end{align*}$$

Let us note here in passing that $\mathbb {P}_{G,H,{\mathcal L},W}$ is simply the pushforward via $f^{-1}$ .

We will refer to these mappings as the lift operator and the projection operator, respectively. When the CPS and window are clear from the context, we will simply write $\mathbb {L}(\mu )$ and $\mathbb {P}(\nu )$ , respectively, instead of $\mathbb {L}_{G,H,{\mathcal L}, W}(\mu )$ and $\mathbb {P}_{G,H,{\mathcal L}, W}(\nu )$ , respectively.

The main results in this paper are that these operators are bijections between the subspaces of Fourier transformable (or cones of positive-definite) measures, and relate their Fourier transforms.

To understand the connection between the Fourier transforms, let us recall the notion of dual CPS. Given a CPS $(G, H, {\mathcal L})$ , we can define

$$ \begin{align*} {\mathcal L}^0 := \{ (\chi, \psi) \in \widehat{G} \times \widehat{H} : \chi(x)\psi(x^{\star}) =1 \, \forall x \in L \} \,. \end{align*} $$

Then, $(\widehat {G}, \widehat {H}, {\mathcal L}^0)$ is a CPS [Reference Baake, Huck and Strungaru5, Reference Moody and Moody28, Reference Moody, Axel, Dénoyer and Gazeau29, Reference Strungaru, Baake and Grimm46]. We will refer to this as the CPS dual to $(G, H, {\mathcal L})$ .

5.1 Strongly admissible functions for CPS

Let us start with the following definition.

Next, given a CPS $(G,H,{\mathcal L})$ , we will denote by ${\mathcal M}_{{\mathcal L}}(G \times H)$ , the space of ${\mathcal L}$ -periodic measures on $G \times H$ . Note that by [Reference Lenz and Richard21, Proposition 6.1]

$$\begin{align*}{\mathcal M}_{{\mathcal L}}(G \times H) \subseteq {\mathcal M}^{\infty}(G \times H) \,. \end{align*}$$

We will see below that given a Fourier transformable measure $\gamma $ with weak Meyer set support, Theorem 4.1 can be used to create a CPS $(G, H={\mathbb R}^d \times H_0, {\mathcal L})$ , an ${\mathcal L}^0$ -periodic measure $\rho (=\widehat {\eta })$ and a strongly admissible function f on $\widehat {H} ={\mathbb R}^d \times \widehat {H_0}$ such that, equation (4.1) yields

$$\begin{align*}\gamma= (\rho)_{f} \,. \end{align*}$$

This motivates us to closely look at the properties of $(\rho )_f$ , for a CPS $(G, H={\mathbb R}^d \times H_0, {\mathcal L}), \rho \in {\mathcal M}_{{\mathcal L}}(G \times H)$ and strongly admissible functions f.

Let us start with the following simple observation which also explains the name “strongly admissible.”

Given a CPS $(G, H={\mathbb R}^d \times H_0, {\mathcal L})$ , a measure $\rho \in {\mathcal M}_{{\mathcal L}}(G \times H)$ , and strongly admissible function f, it is obvious that the function f is admissible for $(G, H, {\mathcal L},\rho )$ in the sense of [Reference Lenz and Richard21, Definition 3.1]. Therefore, by [Reference Lenz and Richard21, Proposition 6.3], we can define a translation bounded measure $\rho _f$ on G via

$$\begin{align*}\rho_f(\phi):= \rho (\phi \otimes f) \qquad \forall \phi \in C_{\mathsf{c}}(G) \,. \end{align*}$$

This measure is strongly almost periodic by [Reference Lenz and Richard21, Theorem 3.1]. In fact, the strong admissibility of f immediately implies that $\rho _f$ is norm almost periodic.

Indeed, let $(G, H={\mathbb R}^d \times H_0, {\mathcal L})$ , let $f=g \otimes \varphi $ be strongly admissible, and let $\rho \in {\mathcal M}_{{\mathcal L}}(G \times H)$ . Pick any compact set $ \mbox {supp}(\varphi ) \subseteq W \subseteq \widehat {H_0}$ , and let $K, K_1 \subseteq G$ be compact sets in G with $K \subseteq K_1^{\circ }$ . Then, a standard computation similar to [Reference Strungaru47, Lemma 5.2] shows that

$$\begin{align*}\| (\rho)_f \|_{K} \leq C \| \varphi \|_{\infty} \| (1+|x|^{2d}) g \|_{\infty} \| \rho \|_{K_1 \times [-\frac{1}{2},\frac{1}{2}]^d \times W } \,, \end{align*}$$

where

$$ \begin{align*} C:= \left( \sum_{n \in {\mathbb Z}^d} \sup_{z \in n+[-\frac{1}{2},\frac{1}{2}]^d } \frac{1}{1+|z|^{2d}} \right) < \infty \,. \end{align*} $$

This immediately gives the following stronger version of [Reference Strungaru47, Lemma 5.2].

5.2 Fourier transform of measures with weak Meyer set support

Fix an arbitrary Meyer set $\Lambda $ and a Fourier transformable measure $\gamma $ with $\mbox {supp}(\gamma ) \subseteq \Lambda $ .

By Theorem 2.10 and the structure theorem of compactly generated groups, there exists a CPS $(G, {\mathbb R}^d \times {\mathbb Z}^m \times \mathbb K, {\mathcal L})$ with compact $\mathbb K$ and a compact $W \subseteq {\mathbb R}^d \times {\mathbb Z}^m \times \mathbb K$ such that

$$\begin{align*}\Lambda \subseteq {\LARGE \curlywedge}(W) \,. \end{align*}$$

By eventually enlarging W, we can assume without loss of generality that

$$\begin{align*}W=W_0 \times F \times \mathbb K \end{align*}$$

for compact $W_0 \subseteq {\mathbb R}^d$ and finite $F \subseteq {\mathbb Z}^m$ .

Set $H_0={\mathbb Z}^m \times \mathbb K$ . It is easy to see that we can find function $\varphi \in C_{\mathsf {c}}^{\infty }({\mathbb R}^d)\cap K_2({\mathbb R}^d)$ and $\psi \in K_2(H_0)$ with the following properties:

• • $\phi := \varphi \otimes \psi \equiv 1$ on W.

• • $\widehat {\psi } \in C_{\mathsf {c}}(\widehat {H_0})$ .

It follows that

(5.1) $$ \begin{align} f:=\check{\phi}= \check{\varphi} \otimes \check{\psi} \end{align} $$

is a strongly admissible function of $\widehat {H} ={\mathbb R}^d \times \widehat {H_0}$ .

Next, define $\eta :=\mathbb {L}_{G, {\mathbb R}^d \times H,{\mathcal L}, W}(\gamma )$ . Then, by Theorem 4.1, $\eta $ is Fourier transformable. Moreover, since $\mbox {supp}(\eta ) \subseteq {\mathcal L}$ , the measure $\rho =\widehat {\eta }$ is ${\mathcal L}^0$ -periodic by [Reference de Lamadrid and Argabright10, Proposition 6.1]. Finally, (4.1) gives

(5.2) $$ \begin{align} \widehat{\gamma}=(\rho)_{f} \,. \end{align} $$

Fact 5.2 then gives the following result.

5.3 Generalized Eberlein decomposition

In this subsection, we show a pseudo-compatibility of the mapping $\rho \to (\rho )_f$ of (5.3), for ${\mathcal L}$ periodic $\rho \in {\mathcal M}_{{\mathcal L}}(G \times H)$ and strongly admissible f, with respect to the Lebesgue decomposition. We explain this, as well as our meaning of “pseudo-compatibility” below.

First, it is easy to see that the map satisfies:

• • if $\rho $ is pure point, then $(\rho )_f$ is pure point;

• • if $\rho $ is absolutely continuous, then $(\rho )_f$ is absolutely continuous;

• • if $\rho $ is singular continuous, then $(\rho )_f$ can have all three spectral components;

and hence does not preserves the Lebesgue decomposition. On another hand, for each $\alpha \in \{\mathsf {pp}, \mathsf {ac}, \mathsf {sc}\}$ , one can defined an operator $P_{\alpha }$ on the space ${\mathcal M}_{{\mathcal L}}^{\infty }(G \times H)$ with the property that for all strongly admissible f and all $\rho \in {\mathcal M}_{{\mathcal L}}^{\infty }(G \times H)$ , we have

(5.3) $$ \begin{align} \left( P_{\alpha} (\rho)\right)_{f}= \left((\rho)_{f}\right)_{\alpha} \,. \end{align} $$

This can be done simply by first showing that

$$\begin{align*}L_{\alpha}(\sum_{j=1}^m c_j \psi_j \otimes \phi_j) := \sum_{j=1}^m c_j \left( \rho_{\phi_j} \right)_{\alpha}(\psi_j) \end{align*}$$

for all $c_1,\ldots ,c_m \in {\mathbb C}, \psi _1,\ldots , \psi _m \in C_{\mathsf {c}}(\widehat {G}), \phi _1,\ldots , \phi _m \in C_{\mathsf {c}}(\widehat {H})$ is well defined, linear, and continuous with respect to the inductive topology. Therefore, $L_{\alpha }$ can be uniquely extended to a measure $P_{\alpha } (\rho )$ , which is ${\mathcal L}$ invariant and satisfies (5.3).

Now, exactly as above, let $\gamma $ be a Fourier transformable measure supported inside a Meyer set $\Lambda $ , and let $(G,H, {\mathcal L}), \eta , \varphi , \phi , \psi $ be as in Section 5.2. Let f be as in (5.1), and let $\rho = \widehat {\eta }$ .

Then, for each $\alpha \in \{\mathsf {pp}, \mathsf {ac}, \mathsf {sc}\}$ , the measure $P_{\alpha } (\rho )$ is the Fourier transform of some measure $\mu $ supported on ${\mathcal L}^0$ [Reference Richard and Strungaru36].

Define

$$\begin{align*}\nu:= \sum_{x \in L} \phi(x^{\star}) \mu(\{ (x,x^{\star}) \}) \delta_x \,. \end{align*}$$

Then, $\mbox {supp}(\nu ) \subseteq {\LARGE \curlywedge }(\mbox {supp}(\phi ))$ and, exactly as in the proof of Theorem 4.1, we get

$$ \begin{align*} \widehat{\nu} &= \left( \widehat{\mu} \right)_{f}= \left( P_{\alpha} (\widehat{\eta}) \right)_{f} =\left((\widehat{\eta})_{h}\right)_{\alpha}= \left( \widehat{\gamma} \right)_{\alpha} \,. \end{align*} $$

Therefore, we get the following corollary.

Corollary 5.4 ([Reference Strungaru47, Theorem 4.1])

Let $\gamma $ be a Fourier transformable measure supported inside a Meyer set $\Lambda $ . Then, there exist a model set $\Gamma \supseteq \Lambda $ and three Fourier transformable measures $\gamma _{\mathsf {s}},\gamma _{\mathsf {0s}}, \gamma _{\mathsf {0a}}$ supported inside $\Gamma $ such that

5.4 Discussion

We have seen in this section that the Fourier transform of a measure $\gamma $ with weak Meyer set support can be describe via (5.2) as the projection in the dual CPS of a ${\mathcal L}^0$ -periodic measure via a strongly admissible function. We used this result to (re)derive properties of $\widehat {\gamma }$ , and we expect that this connection will lead to some new applications in the future. Indeed, while now we know quite a few properties of the Fourier transform of measures with weak Meyer set support [Reference Aujogue2, Reference Strungaru43–Reference Strungaru49], we know much more about fully periodic measures in LCAG (see, for example, [Reference Richard and Strungaru36]). Moreover, the strong admissibility of f is likely to transfer many properties from $\rho $ to $\rho _f$ . It is also worth pointing out that, while the strong admissibility of f was sufficient to derive the conclusions in this section, in fact f can be chosen of the form

$$\begin{align*}f := g \otimes P \otimes \psi :{\mathbb R}^d \times \mathbb T^m \times \widehat{\mathbb K} \to {\mathbb C} \,, \end{align*}$$

where $g =\hat {\varphi } \in {\mathcal S}({\mathbb R}^d)$ is the Fourier transform of some $\varphi \in C_{\mathsf {c}}^{\infty }({\mathbb R}^d)$ ; P is a trigonometric polynomial, that is, a sum of characters, that is, $P= \sum _{j=1}^m \chi _j$ for some $\chi _1,\ldots , \chi _j \in \widehat {\mathbb T^m}$ and $\psi \in C_{\mathsf {c}}(\widehat {\mathbb K})$ is the characteristic function of $\{ 0 \}$ . These properties are much stronger than strong admissibility, and have the potential to lead to nice applications in the future.